reserve V for RealLinearSpace,
  u,u1,u2,v,v1,v2,w,w1,x,y for VECTOR of V,
  a,a1,a2,b,b1,b2,c1,c2,n,k1,k2 for Real;

theorem
  Gen x,y & u,v,u1,v1 are_COrtm_wrt x,y & u,v,u1,w are_COrtm_wrt x,y
  implies u,v,v1,w are_COrtm_wrt x,y or u,v,w,v1 are_COrtm_wrt x,y
proof
  assume that
A1: Gen x,y and
A2: u,v,u1,v1 are_COrtm_wrt x,y and
A3: u,v,u1,w are_COrtm_wrt x,y;
A4: Ortm(x,y,u),Ortm(x,y,v) // u1,v1 by A2;
A5: Ortm(x,y,u),Ortm(x,y,v) // u1,w by A3;
A6: now
    assume that
A7: u<>v and
A8: u1<>v1;
    u1,v1 // u1,w by A1,A4,A5,A7,Th6,ANALOAF:11;
    then
A9: u1,v1 // v1,w or u1,w // w,v1 by ANALOAF:14;
    now
      assume
A10:  u1<>w;
A11:  u1,v1 // Ortm(x,y,u),Ortm(x,y,v) by A4,ANALOAF:12;
      u1,w // Ortm(x,y,u),Ortm(x,y,v) by A5,ANALOAF:12;
      then Ortm(x,y,u),Ortm(x,y,v) // v1,w or
      Ortm(x,y,u),Ortm(x,y,v) // w,v1 by A8,A9,A10,A11,ANALOAF:11;
      hence thesis;
    end;
    hence thesis by A2;
  end;
  u=v implies thesis by ANALOAF:9;
  hence thesis by A3,A6;
end;
